Formula:KLS:09.15:02

<< Formula:KLS:09.15:01

formula in Hermite

Formula:KLS:09.15:03 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}^{-x^{2}}}H_{m}\left(x\right)H_{n}\left(x\right)\,dx=2^{n}n!\,\delta_{m,n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : Hermite polynomial ${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : http://dlmf.nist.gov/18.3#T1.t1.r28
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.15 of KLS.

URL links

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<< Formula:KLS:09.15:01

formula in Hermite

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